The definition of an Anosov map is the same as for an Anosov diffeomorphism except that instead of being a diffeomorphism,
it is a map. In particular, an Anosov map is a map f of a manifold to itself such that the tangent
bundle of
is hyperbolic with respect to .
A trivial example is to map all of to a single point of . Here, the eigenvalues are all zero. A less trivial example
is an expanding map on the circle , e.g., , where is identified with the real numbers (mod 1). Here, all the
eigenvalues equal 2 (i.e., the eigenvalue at each point of ). Note that this map is not a diffeomorphism
because ,
so it has no inverse.
A nontrivial example is formed by taking Arnold's cat map on the 2-torus ,
and crossing it with an expanding map on to form an Anosov map on the 3-torus , where denotes the Cartesian
product. In other words,
Anosov, D. "Roughness of Geodesic Flows on Compact Riemannian Manifolds of Negative Curvature." Dokl. Akad. Nauk SSSR145,
707-709, 1962. English translation in Soviet Math. Dokl.3, 1068-1069,
1962.Anosov, D. "Ergodic Properties of Geodesic Flows on Closed
Riemannian Manifolds of Negative Curvature." Dokl. Akad. Nauk SSSR151,
1250-1252, 1963. English translated in Soviet Math. Dokl.4, 1153-1156,
1963.Lichtenberg, A. J. and Lieberman, M. A. Regular
and Chaotic Dynamics, 2nd ed. New York: Springer-Verlag, pp. 305-307,
1992.Sondow, J. "Fixed Points of Anosov Maps of Certain Manifolds."
Proc. Amer. Math. Soc.61, 381-384, 1976.
map f of a manifold 
to itself such that the tangent
bundle of 
is hyperbolic with respect to 
.
to a single point of 
. Here, the eigenvalues are all zero. A less trivial example
is an expanding map on the circle 
, e.g., 
, where 
is identified with the real numbers (mod 1). Here, all the
eigenvalues equal 2 (i.e., the eigenvalue at each point of 
). Note that this map is not a diffeomorphism
because 
,
so it has no inverse.
,
and crossing it with an expanding map on 
to form an Anosov map on the 3-torus 
, where 
denotes the Cartesian
product. In other words,
![[x_(n+1); y_(n+1); z_(n+1)]=[1 1 0; 1 2 0; 0 0 2][x_n; y_n; z_n] (mod 1).](/images/equations/AnosovMap/NumberedEquation1.svg)
